Browsing by Author "Bloch, Anthony M."
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Item Dissipation Induced Instabilities(1992) Bloch, Anthony M.; Krishnaprasad, Perinkulam S.; Marsden, Jerrold E.; Ratiu, Tudor S.; ISRThe main goal of this paper is to prove that if the energy- momentum (or energy-Casimir) method predicts formal instability of a relative equilibrium in a Hamiltonian system with symmetry, then with the addition of dissipation, the relative equilibrium becomes spectrally and hence linearly and nonlinearly unstable. The energy-momentum method assumes that one is in the context of a mechanical system with a given symmetry group. Our result assumes that the dissipation chosen does not destroy the conservation law associated with the given symmetry group -- thus, we consider internal dissipation. Our result also includes the special case of systems with no symmetry and ordinary equilibria. Our result is proved by combining the techniques of Chetaev, who proved instability theorems using a special Chetaev- Lyapunov function, those of Hahn, which enable one to strengthen the Chetaev results from Lyapunov instability to spectral instability. Our main achievement is to strengthen these results to the context of the block diagonalization version of the energy momentum method given by Lewis. Marsden, Posbergh, and Simo. However, we also give the eigenvalue movement formulae of Krein, MacKay and others both in general and adapted to the context of the normal form of the linearized equations given by the block diagoanl form as provided by the energy-momentum method. A number of specific examples, such as the rigid body with internal rotors, are provided to illustrate the results.Item Stabilization of Rigid Body Dynamics by Internal and External Torques(1990) Bloch, Anthony M.; Krishnaprasad, Perinkulam S.; Marsden, Jerrold E.; Sanchez de Alvarez, G.; ISRIn this paper we discuss the stabilization of the rigid body dynamics by external torques (gas jets) and internal torques (momentum wheels). We compare the stabilizing quadratic quadratic feedback law for a single external torque recently analyzed in Bloch and Marsden [1989b,c] with quadratic feedback torques for internal rotors. We show that with such torques, the equations for the rigid body with momentum wheels are Hamiltonian with respect to a Lie-Poisson bracket structure. Further, these equations are shown to generalize the dual-spin equations analyzed by Krishnaprasad [1985] and Sanchez de Alvarez [1986]. We establish stabilization with a single rotor by using the energy-Casimir method. We also show how to realize the external torque feedback equations using internal torques. Finally, extending some work of Montgomery [1990], we derive a formula for the attitude drift for the rigid body-rotor system when it is perturbed away from stable equilibrium and we indicate how to compensate for this.